Learn Turkish: Bir şüpheliyle yalnız kalmam. - I do not stay alone with a suspect.

This is the irregular 1st person singular of the aorist (simple present) negative. Normally, the aorist uses an -r- (like in kalırım, "I stay"). But in the negative for "I" and "we", that -r- drops out completely! We just use the verb root kal-, the negative suffix -ma-, and the personal ending -m to get kalmam.

Where does the -yle come from in şüpheliyle?

It is the attached suffix form of ile, which means "with". Because şüpheli (suspect) ends in a vowel, we have to insert a buffer -y- before attaching the suffix -le (which follows vowel harmony). So, şüpheli + -y- + -le gives us "with a suspect".

I learned that şüpheli is an adjective meaning "suspicious". Why is it translated as a noun ("a suspect") here?

In Turkish, it is extremely common to use adjectives directly as nouns to describe people. Just like hasta can mean "sick" or "a sick patient", şüpheli can mean "suspicious" or "a suspect" depending on the context.

Could yalnız kalmam also mean "I am not alone"?